First pick one of the five elements to be fixed. For each such choice, derange the remaining four, using the standard advanced PIE formula. We get \({5 \choose 1}\left( 4! - \left[{4 \choose 1}3! - {4 \choose 2}2! + {4 \choose 3} 1! - {4 \choose 4} 0!\right] \right)\) permutations.
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